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Abstract

Dual photoelastic modulator polarimeter systems are widely used for the measurement of light beam polarization, most often described by Stokes vectors, that carry information about an interrogated sample. The sample polarization properties can be described more thoroughly through its Mueller matrix, which can be derived from judiciously chosen input polarization Stokes vectors and correspondingly measured output Stokes vectors. However, several sources of error complicate the construction of a Mueller matrix from the measured Stokes vectors. Here we present a general formalism to examine these sources of error and their effects on the derived Mueller matrix, and identify the optimal input polarization states to minimize their effects in a dual photoelastic modulator polarimeter configuration. The input Stokes vector states leading to the most robust Mueller matrix determination are shown to form Platonic solids in the Poincaré sphere space; we also identify the optimal 3D orientation of these solids for error minimization.

Figures (6)

The dual PEM Stokes polarimeter schematic. Panel (a) provides a top view of the system, which is comprised of a movable PSG to illuminate the sample and dual photoelastic modulators followed by a linear polarizer and a photodetector, which form the PSA. The two lock-in amplifiers recover the polarization parameters Q, U, and V using the PEM modulation frequencies as references. Panel (b) depicts the PSA as seen from the photodetector. The fast axis of PEM 1 is at an angle α = 45° above the horizontal (laboratory frame), and the fast axis of PEM 2 is parallel to the optical table. The linear polarizer’s transmission axis is at an angle β = 22.5° above the horizontal.

The five Platonic solids inscribed in the Poincaré sphere, where the equator represents linear polarization and the poles represent circular polarization. The three great circles on each sphere represent the areas where Q, U, or V are zero. When the input Stokes vectors form the vertices of any of these shapes, Eqs. (21)–(23) are satisfied, and so the error-sensitivity of the determined Mueller matrix will be minimized. In (a) the n = 4 input Stokes vectors form a tetrahedron in polarization space (
Media 1), in (b) where n = 6, they form an octahedron (
Media 2), in (c) where n = 8, they form a cube (
Media 3), in (d) where n = 12, they form an icosahedron (
Media 4), and finally, in (e) where n = 20, they form a dodecahedron (
Media 5).

(
Media 6). The Poincaré sphere coloured according to the likelihood of phase errors in each region, where red represents a high probability and green represents a low probability. As in Fig. 2, the great circles in black represent the areas where the polarization parameters are 0. As Stokes vectors approach these areas (i.e. they enter the red/yellow zones) they become increasingly prone to phase errors. Here, the cubic configuration of Fig. 2(c) (
Media 3) has been inscribed in the sphere to show that the Stokes vectors forming its vertices are all minimally prone to phase errors.

The probability of a simulated phase inversions as a function of q, u, or v. The likelihood of a phase error approaches 0.5 when the polarization parameters are near 0, and it drops quickly to ∼ 0 as said parameters move towards ±1.

(
Media 7).The Mueller errors resulting from ∼ 104 randomly chosen sets of n Stokes vectors, evenly distributed over 4 ≤ n ≤ 20. Each blue dot represents one simulated Mueller matrix determination, and 〈δM〉 on the vertical axis shows the RMS error in said matrix. On the horizontal plane, n and ||(in)+|| serve to describe and quantify each configuration. The red dot surrounded by the black circle at the bottom of the plot shows the simulation result corresponding to the optimal cubic configuration in Fig. 3 (
Media 6).